Manual for the IEEE CEC 2014 Competition Interdependent Components

Manual for the IEEE CEC 2014 Competition
Optimisation of Problems with Multiple
Interdependent Components
Sergey Polyakovskiy, Mohammad Reza Bonyadi, Markus Wagner,
Zbigniew Michalewicz, Frank Neumann
School of Computer Science, The University of Adelaide
Abstract
Real-world optimization problems often consist of several NP-hard optimization problems that interact with each other. The goal of this manual
is to describe a benchmark suite that promotes a research of the interaction between problems and their mutual influence. We establish a comprehensive benchmark suite for the traveling thief problem (TTP) which
combines the traveling salesman problem and the knapsack problem. Our
benchmark suite builds on common benchmarks for the two sub-problems
which grant a basis to examine the potential hardness imposed by combining the two classical problems.
The official website of the IEEE CEC 2014 Competition Optimisation of Problems with Multiple Interdependent Components is http://
cs.adelaide.edu.au/~optlog/CEC2014Comp/. If you have any questions,
please do not hesitate to contact the organizers.
1
Introduction
Real-world optimization problems usually consist of several problems that interact with each other. In order to solve such problems it is important to
understand and deal with these interactions. So far, the research literature is
lacking systematic approaches for dealing with such interdependent problems.
In this paper, we consider the Traveling Thief Problem (TTP) [4] which is
a combination of two of the most prominent combinatorial optimization problems, namely the traveling salesperson problem (TSP) and the knapsack problem (KP). Both problems have been considered in numerous theoretical and
experimental studies, and very effective solvers are known that perform well on
a variety of benchmarks.
We present a benchmark suite for TTP which is based on benchmark instances for the TSP and KP. The aim of this benchmark suite is to give researchers the opportunities
• to study TTP as a combination of TSP and KP,
1
• to compare the problem instances, and
• to provide algorithms that can effectively solve problems with interdependencies.
We systematically construct the benchmarks so that the resulting instances
cover a wide range of features: from few cities with few items and small knapsacks to many cities with many items and large knapsacks. While we are confident that smaller instances can soon be solved to optimality, the larger ones
most likely remain unsolved for the years to come.
It is important to note that the TTP is unlike many capacitated vehiclerouting problems in the area of Green Logistics (see, e.g., the survey article [7]).
For example, the fuel consumption based on load or geographical features is
considered in [5, 6] and the problem for several vehicles is solved using integer
programming. In our case, we add to the routing problem not only a loaddependent feature, but also the NP-hard optimization problem of deciding which
items are to be packed.
The remainder of this manual is organized as follows. First, we define the
traveling thief problem in Section 2. In Section 3 we describe the systematic
construction of the benchmark set.
2
Traveling Thief Problem
The Traveling Thief Problem is defined as follows. Given is a set of cities
N = {1, . . . , n} for which a distance dij , i, j ∈ N between any pair of cities is
known. Every city i but the first contains a set of items Mi = {1, . . . , mi }. Each
item k positioned in the city i is characterized by its value pik and weight wik ,
Iik ∼ (pik , wik ). The thief must visit each of the cities exactly once starting
from the first city and returning back to it in the end. Any item may be picked
up into the knapsack in any city until the total weight of collected items does
not exceed the maximum possible weight W . A renting rate R is to be paid
per each time unit being on a way. υmax and υmin denote the maximal and
minimum speeds that the thief can move, respectively. The goal is to find a
tour of the maximal profit.
Let yik ∈ {0, 1} be a binary variable equal to one when the item k is picked
up in the city i. In addition, let Wi denote the total weight of collected items
when the thief leaves the city i. Therefore, the objective function for a tour
Π = (x1 , . . . , xn ), xi ∈ N and a packing plan P = (y21 , . . . , ynmi ) takes the
following form:
Z(Π, P )
=
mi
n X
X
pik yik
i=1 k=1
n−1
−R
X
dxi xi+1
dxn x1
+
υmax − νWxn
υ
− νWxi
i=1 max
2
!
−υmin
where ν = υmaxW
is a constant value. The minuend is the sum over all
packed items’ profits and the subtrahend is the amount that the thief pays for
the knapsack’s rent equal to the total traveling time along Π multiplied by R.
A numeric example of the TTP problem is provided via the graph given in
Figure 1. Each node but the first has an assigned set of items, e.g. node 2 is
associated with item I21 of profit p21 = 20 and weight w21 = 2, and with item
I22 of profit p22 = 30 and weight w22 = 3. Let’s assume that the maximum
weight W = 3, the renting rate R = 1 and υmax and υmin are set as 1 and 0.1,
respectively. Then the optimum objective value Z(Π, P ) = 50 for Π = (1, 2, 4, 3)
and P = (0, 0, 0, 1, 1, 0). Specifically, the thief collects no items traveling from
city 1 to city 3 via cities 2 and 4. Therefore, this part of the tour costs 15.
Only in the city 3 items I32 and I33 are picked up, that gives total profit of
80. However, on the way from city 3 back to city 1 the thief’s knapsack has
weight of 2. In fact, it reduces the speed and results in increased cost of 15.
Consequently, the final objective value is Z(Π, P ) = 80 − 15 − 15 = 50.
Figure 1: Illustrative Example
3
Benchmark Set
Herein we propose a new benchmark of instances in order to promote a correct comparison among different approaches to the TTP problem. It is mainly
motivated by a desire to (1) have a combination of well-known test suites for
each sub-problem, and (2) a correct adjustment of constant variables involved
in the objective function. Specifically, from an experimental point of view, it is
important to keep a balance between two components of the problem. It means
that the near-optimal solution of one sub-problem must not dominate over the
optimal solution of another sub-problem. Therefore, solving the shortest tour
problem to optimality must not make the knapsack packing aspect negligible.
Vice versa, the most profitable loading plan must not reduce the importance of
a shorter tour.
To handle this task the most known set of test instances for each sub-problem
was considered. The TSP library collected by Reinelt [2, 9] is to be the main
departure point when one is going to evaluate his/her approach’s performance
on the TSP. It consists of over 100 instances with sizes ranging from 14 to 85,900
3
cities and is derived from industrial applications and geographic problems. Since
its creation in 1990, an extensive investigation has been carried out and now all
the symmetric TSP library instances are solved to optimality.
The knapsack problem was thoroughly studied by Martello, Pisinger and
Toth [8]. Their work deals with the 0-1 knapsack and proposes the exact dynamic programming algorithm called COMBO to tackle it. Furthermore, they
propose and investigate a test benchmark wealthy in ways to generate KP data.
COMBO adopts a set of valid inequalities along with a new initial core problem
that allows it to solve instances with up to 10,000 items in less than 0.2 seconds.
The authors generated fifteen different types of KP varying weights and profits
of items to investigate which aspects result in challenges for KP solvers in general and for their approach in particular. From their work, we pick several of
the hardest cases and combine them with instances of the TSP library.
To generate our TTP instances we first consider the instances of the TSP
library. First, we select the 81 instances that possess “EUC 2D” or “CEIL 2D”
marks for their edge weight type tag. Despite the mark derived from the TSP
library we assume that over the entire TTP benchmark suite all distances between the cities are integer values rounded up to the next integer. The resulting
set has instances with the number of cities ranging from 51 and up to 85,900.
For each TSP instance, we generate a knapsack component using the code
from [1]. Three ways to generate a KP data are selected respecting the evaluated hardness for solving and the recommendations reported in [8]. Each way
results in a unique knapsack type. In the following, we distinguish between
uncorrelated, uncorrelated with similar weights and bounded strongly correlated
types.
For the uncorrelated type a weight wik and
pik of item k are uni a profit
formly distributed random integer values in 1, 103 . Loose correlation or no
correlation among the profit and weight of each item makes an instance easy to
solve to optimality even for large-sized problems [8]. However, this may be not
straightforward for the TTP problem and a behavior of algorithms stays of a
particular interest for research.
Addressing the uncorrelated type with similar weights
we employ
for wik
and pik integer values uniformly distributed within 103 , 103 + 10 and 1, 103
respectively. Producing items with similar weights complicates KP by increasing
the core problem’s size. It has already been shown that this type of 0-1 knapsack
instances is more time consuming to be solved [8]. Consequently, having all items
of almost equal weight but definitely different profit may require an additional
effort to develop an algorithm which intelligently deal with this aspect in TTP,
too.
Oppositely to the uncorrelated case, strongly correlated problems are typically very difficult. The bounded strongly correlated type adopts the bounded
knapsack problem where multiple copies of an item are allowed to be selected
complying with a maximum number restriction. As previously shown, its transformation to a 0-1 knapsack problem complicates the problem [8]. It significantly increases the computation time while it forces the cardinality constraints
in COMBO to loose their effect. This indicates that specialized algorithms for the
4
bounded version should be developed.
To set up the initial bounded instance,
we choose the interval 1, 103 for an item’s integer weight wik uniformly distributed generation and we set pik based on this as wik + 100.
To diversify the size of the knapsack component, we distinguish between four
values of an item factor F ∈ {1, 3, 5, 10}. In our benchmarks, F describes how
many items per city are available. We use the same number of items for every
city setting F = mi for any i ∈ N , i ≥ 2. As assumed in the problem statement,
the first city does not contain any item. Therefore, for a number of n cities we
produce totally F × (n − 1) items.
Next, for each TSP library instance, knapsack type and item factor we generate 10 instances. Each instance falls into its capacity category C ∈ {1, . . . , 10}
according to the instance number. The capacity category value C results in a
n X
F
X
C
wik . Effectively,
unique maximum Knapsack weight value of W = 11
i=1 k=1
C is the factor by which the capacity of the smallest knapsack (in the set of 10)
is multiplied.
As noted above, we try to manage the potential objective value by searching
for a balance between TSP and KP components through the constant variables. Specifically, we set the same values for υmax = 1 and υmin = 0.1 over
all instances. However, each instance gets its individual value R defined as
Z (P OP T )
R = T IM E(Πlinkern ,P OP T ) , where Z P OP T corresponds to the optimal profit
of the KP component and T IM E Πlinkern , P OP T denotes the total traveling
time along the near-optimal TSP tour Πlinkern obtained via the Chained LinKernighan heuristic [3]1 while picking the items according to the optimal KP
component’s solution P OP T . Such selection of R guarantees the existence of at
least one TTP solution with zero objective value.
Consequently, our final benchmark set based on 81 TSP instances, three KP
types, four item factors and ten capacity categories contains 81 × 3 × 4 × 10 =
9, 720 different TTP instances in total.2
4
Concluding Remarks
The official website of the IEEE CEC 2014 Competition Optimisation of Problems with Multiple Interdependent Components is http://cs.adelaide.edu.
au/~optlog/CEC2014Comp/. If you have any questions, please do not hesitate
to contact the organizers.
1 As
available at http://www.tsp.gatech.edu/concorde/downloads/downloads.htm
instances and implementations of the objective function are available online: (link
removed for the review process)
2 All
5
References
[1] Advanced generator for 0-1 Knapsack Problems. See http://www.diku.dk/
~pisinger/codes.html.
[2] TSP Test Data. See http://comopt.ifi.uni-heidelberg.de/software/
TSPLIB95/index.html.
[3] D. Applegate, W. J. Cook, and A. Rohe. Chained lin-kernighan for large
traveling salesman problems. INFORMS Journal on Computing, 15:82–92,
2003.
[4] M. R. Bonyadi, Z. Michalewicz, and L. Barone. The travelling thief problem: The first step in the transition from theoretical problems to realistic
problems. In IEEE Congress on Evolutionary Computation, pp. 1037–1044.
IEEE, 2013.
[5] I. Kara, B. Y. Kara, and M. K. Yetis. Energy minimizing vehicle routing
problem. In A. Dress, Y. Xu, and B. Zhu, editors, Combinatorial Optimization and Applications, Vol. 4616 of Lecture Notes in Computer Science, pp.
62–71. Springer Berlin Heidelberg, 2007.
[6] I. Kucukoglu, S. Ene, A. Aksoy, and N. Ozturk. Green capacitated vehicle routing problem fuel consumption optimization model. Computational
Engineering Research, 3:16–23.
[7] C. Lin, K. Choy, G. Ho, S. Chung, and H. Lam. Survey of green vehicle
routing problem: Past and future trends. Expert Systems with Applications,
41:1118 – 1138, 2014.
[8] S. Martello, D. Pisinger, and P. Toth. Dynamic programming and strong
bounds for the 0-1 knapsack problem. Manage. Sci., 45:414–424, 1999.
[9] G. Reinelt. TSPLIB - A Traveling Salesman Problem Library. ORSA Journal on Computing, 3:376–384, 1991.
6